Solve 1+sqrt{x+15}-4=4x-3

$$1+ \sqrt{ x+15 } -4 = 4x-3$$

Solve for x

$$x=1$$

Solution Steps

Subtract 1-4 from both sides of the equation.

$$\sqrt{x+15}=4x-3-\left(1-4\right)$$

Subtract 4 from 1 to get -3.

$$\sqrt{x+15}=4x-3-\left(-3\right)$$

Multiply -1 and -3 to get 3.

$$\sqrt{x+15}=4x-3+3$$

Add -3 and 3 to get 0.

$$\sqrt{x+15}=4x$$

Square both sides of the equation.

$$\left(\sqrt{x+15}\right)^{2}=\left(4x\right)^{2}$$

Calculate \sqrt{x+15} to the power of 2 and get x+15.

$$x+15=\left(4x\right)^{2}$$

Expand \left(4x\right)^{2}.

$$x+15=4^{2}x^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x+15=16x^{2}$$

Subtract 16x^{2} from both sides.

$$x+15-16x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-16x^{2}+x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -16x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=1 ab=-16\times 15=-240$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -240.

$$-1,240 -2,120 -3,80 -4,60 -5,48 -6,40 -8,30 -10,24 -12,20 -15,16$$

Calculate the sum for each pair.

$$-1+240=239 -2+120=118 -3+80=77 -4+60=56 -5+48=43 -6+40=34 -8+30=22 -10+24=14 -12+20=8 -15+16=1$$

The solution is the pair that gives sum 1.

$$a=16 b=-15$$

Rewrite -16x^{2}+x+15 as \left(-16x^{2}+16x\right)+\left(-15x+15\right).

$$\left(-16x^{2}+16x\right)+\left(-15x+15\right)$$

Factor out 16x in the first and 15 in the second group.

$$16x\left(-x+1\right)+15\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(16x+15\right)$$

To find equation solutions, solve -x+1=0 and 16x+15=0.

$$x=1 x=-\frac{15}{16}$$

Substitute 1 for x in the equation 1+\sqrt{x+15}-4=4x-3.

$$1+\sqrt{1+15}-4=4\times 1-3$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

Substitute -\frac{15}{16} for x in the equation 1+\sqrt{x+15}-4=4x-3.

$$1+\sqrt{-\frac{15}{16}+15}-4=4\left(-\frac{15}{16}\right)-3$$

Simplify. The value x=-\frac{15}{16} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{3}{4}=-\frac{27}{4}$$

Equation \sqrt{x+15}=4x has a unique solution.

$$x=1$$